Problem 12 - Precalculus : Building Concepts and Connections

Question

Write each of the given vectors in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$\mathbf{w}=\left\langle-\frac{2}{5}, \frac{1}{6}\right\rangle$$

Step-by-Step Solution

Answer

\(-\frac{2}{5}\mathbf{i} + \frac{1}{6}\mathbf{j}\)

Step 1: Identify the x and y components of the given vector

For the given vector \(\mathbf{w}=\left\langle-\frac{2}{5}, \frac{1}{6}\right\rangle\), the x-component is \(-\frac{2}{5}\) and the y-component is \(\frac{1}{6}\)

Step 2: Express the vector in terms of unit vectors

Substitute the x and y components into the standard form \(v_x \mathbf{i} + v_y \mathbf{j}\), this yields \(-\frac{2}{5}\mathbf{i} + \frac{1}{6}\mathbf{j}\)

Key Concepts

Vector Components

To fully grasp the concept of vectors in precalculus, one must first understand vector components. Virtually every vector can be broken down into parts that run along the x and y axis, known respectively as the vector's x-component and y-component. Think of it like deciphering a treasure map: the x-component tells you how far to walk east or west, and the y-component signals the distance north or south.

For example, the vector \( \mathbf{w} \), with components \(\left\langle -\frac{2}{5}, \frac{1}{6} \right\rangle\), consists of two parts: \( -\frac{2}{5} \) moving along the x-axis and \( \frac{1}{6} \) along the y-axis. This information guides you to plot the vector's direction and magnitude precisely on a coordinate plane.

Expressing Vectors

When we communicate direction and magnitude in vector form, it's like providing a crisp, clear set of instructions for drawing an arrow on a graph. Referring back to our treasure map analogy, this arrow shows the exact path and distance to the treasure. Expressing a vector in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) streamlines this communication process. The unit vector \( \mathbf{i} \) is an arrow pointing directly to the right, representing a movement of one unit on the x-axis, while \( \mathbf{j} \) indicates one unit up on the y-axis.

In the expression \( -\frac{2}{5}\mathbf{i} + \frac{1}{6}\mathbf{j} \), each component clearly exhibits how many steps right/left and up/down are taken. Thus, \( \mathbf{w} \) is painted as a vector moving \( -\frac{2}{5} \) units on the x-axis and \( \frac{1}{6} \) units on the y-axis.

Standard Form of Vectors

The standard form of a vector blends simplicity with precision, allowing anyone to visualize a vector's direction and length instantly. This form typically writes a vector as \( v_x \mathbf{i} + v_y \mathbf{j} \) where \( v_x \) and \( v_y \) are the scalar components. Returning to our previous vector \( \mathbf{w} = \left\langle -\frac{2}{5}, \frac{1}{6} \right\rangle \), it translates to the x-component (\(\ -\frac{2}{5}\)) times the unit vector in the x-direction \( \mathbf{i} \) summed with the y-component (\(\ \frac{1}{6}\)) times the unit vector in the y-direction \( \mathbf{j} \).

This elegant notation allows physicists, engineers, and mathematicians to effortlessly convey precise movements through space. When spotted, the standard form alerts the reader to visualize a vector's component contribution to its overall direction and magnitude.